From Rotation to Discontinuity: Hovering Orbits in Perturbed Hamiltonian Impact Systems

Hamiltonian impact systems describe mechanical systems that evolve smoothly inside a domain and undergo elastic reflections at its boundary, modeling, for example, potentials with sharp local gradients. I will briefly review some of the main results on these systems. Then, I will discuss recent progress on perturbed pseudo-integrable Hamiltonian impact systems, motivated by a simple mechanical model whose unperturbed dynamics are pseudo-integrable (notably, its quantum counterpart has also been studied, see Phys. Rev. E 107, 054221 (2023)). In this model, small perturbations create a positive-measure set of hovering orbits, although such orbits are absent in the unperturbed dynamics. This provides a striking example of how discontinuities and impacts can generate qualitatively new behavior in near-pseudo-integrable Hamiltonian systems.

Joint work with I. Pazi and A. Zobova, see https://arxiv.org/abs/2509.24688 (to appear in SIAM journal on Applied Dynamical Systems) and https://arxiv.org/abs/2605.27987v1 .